Here, \(\mathcal {L}\)  is the action of the symmetry groups, including the gauge group \(G\)  and the isometry group of \(X\)  containing the maximal torus \(H=T^2\) . We denote these collectively by \(L\) . From (REF ) we see that \((\phi ,\psi )\)  and \((\tilde{\psi },\tilde{\phi })\)  form multiplets with \(\mathcal {Q}^2=\mathcal {L}\) , i.e. \(\mathcal {Q}\)  can be viewed as an equivariant differential. In order to perform localization, the action is deformed by a \(\mathcal {Q}\) -exact term \(\delta S=t\mathcal {Q}V\)  (\(t\in \mathbb {R}\) ) such that \(\mathcal {Q}^2 V=0\)  (i.e. \(V\)  is \(\mathcal {L}\) -invariant) and \(\mathcal {Q}V\)  is positive semi-definite. By the standard argument, this deformation is independent of the value of \(t\)  and we choose \(t\rightarrow \infty \) . The path integral then localises to field configurations such that \(\mathcal {Q}V=0\) . The one-loop part is obtained by expanding \(\delta S\)  around these configurations to second order, producing a quadratic (Gaussian) term which can be integrated to the one-loop determinant [1]}
\(\frac{\det ^{1/2}|_{\operatorname{coker}\eth }\mathcal {L}}{\det ^{1/2}|_{\ker \eth }\mathcal {L}}=\operatorname{sdet}^{1/2}|_{H^\bullet (\eth )}\mathcal {L},\) 
